research papers
Acta Crystallographica Section A
Foundations of
Crystallography
The charge density of urea from synchrotron
diffraction data
ISSN 0108-7673
Received 28 February 2004
Accepted 21 June 2004
Henrik Birkedal,a*³2Dennis Madsen,b*§4 Ragnvald H. Mathiesen,c}6
Kenneth Knudsen,d³³8Hans-Peter Weber,d,a Philip Pattisond,a and Dieter
Schwarzenbacha
a
Laboratory of Crystallography, Swiss Federal Institute of Technology, BSP, CH-1015 Lausanne,
Switzerland, bEuropean Synchrotron Radiation Facility, Rue Jules Horowitz 6, BP 220, F-38043
Grenoble CEDEX, France, cInstitutt for Fysikk, NTNU, N-7491 Trondheim, Norway, and dSwiss±
Norwegian Beam Lines at ESRF, Rue Jules Horowitz 6, BP 220, F-38043 Grenoble CEDEX, France.
Correspondence e-mail: [email protected], [email protected]
# 2004 International Union of Crystallography
Printed in Great Britain ± all rights reserved
The charge density of urea is studied using very high precision single-crystal
synchrotron-radiation diffraction data collected at the Swiss±Norwegian Beam
Ê ÿ1 in sin = is obtained
Lines at ESRF. An unprecedented resolution of 1.44 A
at 123 K. The optimization of the experiment for charge-density studies is
discussed. The high precision of the data allowed the re®nement of a multipole
model extending to hexadecapoles and quadrupoles on the heavy and H atoms,
respectively, as well as a liberal treatment of radial functions. The topological
properties of the resulting electron density are analysed and compared with
earlier experimental results as well as with periodic Hartree±Fock calculations.
The properties of the strongly polarized CÐO bond agree with trends derived
from previous experimental results while the ab initio calculations differ
signi®cantly. The results indicate that the description of the CÐO bond requires
more ¯exible basis sets in the theoretical calculations. The calculated integrated
atomic charges are much larger than the observed ones. It is suggested that the
present experimental results provide new target values for validation of future
ab initio calculations. The molecular dipole moment derived from the integrated
atomic properties is the same as the one obtained from the multipole model
even though the individual atomic contributions differ. Comparison with
literature data for urea in solution and the gas phase yields a dipole
enhancement in the solid of about 1.5 D. The thermal expansion of urea is
determined using synchrotron powder diffraction data. With decreasing
temperature, an increasing anisotropic strain is observed.
1. Introduction
Urea has attracted signi®cant interest in the charge-density
community because it is a simple, chemically interesting, noncentrosymmetric structure (Swaminathan, Craven, Spackman
& Stewart, 1984; Spackman & Byrom, 1997; Zavodnik et al.,
1999; Spackman et al., 1999; de Vries et al., 2000). In particular,
it has served as a model structure for studies of phase retrieval
in non-centrosymmetric structures. Urea has been extensively
studied with many techniques. Single-crystal neutron diffraction studies have been performed by Swaminathan, Craven &
³ Present address: Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus, Denmark.
§ Present address: Novo Nordisk A/S, Department of Scienti®c Computing,
Novo Nordisk Park, A2P, DK-2760 MaÊlùv, Denmark
} Present address: SINTEF, Applied Physics Department, N-7465 Trondheim,
Norway.
³³ Present address: Institute for Energy Technology, Physics Department,
N-2007 Kjeller, Norway.
Acta Cryst. (2004). A60, 371±381
McMullan (1984) and in several previous works cited therein.
Swaminathan, Craven & McMullan (1984) collected full data
sets at 12, 60 and 123 K with additional determination of
lattice parameters at 30, 90, 150 and 173 K. Hereafter we refer
to this study as SCM. Two earlier experimental charge-density
studies have been published. Swaminathan, Craven,
Spackman & Stewart (1984), hereafter SCSS, measured at
Ê ÿ1 in
123 K the sector hkl, h k, to a maximum of 1.15 A
sin =. Zavodnik et al. (1999), hereafter ZSTVF, measured a
more complete data set to the same resolution but at 148 K.
They then used scaled values of the neutron H-atom parameters of SCM in the X-ray re®nements. They obtained
displacement parameters of the heavy atoms from a highorder re®nement and used only the 145 re¯ections at sin = Ê ÿ1 for the multipole re®nement.
0.7 A
Urea was the ®rst molecular crystal to be studied with fully
periodic Hartree±Fock calculations by Dovesi et al. (1990)
using the CRYSTAL program. This study was followed by the
DOI: 10.1107/S0108767304015120
371
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®rst application of the quantum theory of atoms in molecules
(AIM) on a molecular crystal by Gatti et al. (1994), using
CRYSTAL in combination with Gatti's AIM program
TOPOND. The ®rst study used the 6-21G(d,p) basis set while
the latter one employed 6-31G(d,p). It should be mentioned
that such relatively small basis sets perform better in periodic
than in molecular calculations (Pisani et al., 1988; Pisani,
1996). The AIM results were extended by Gatti & Cargnoni
(1997). Gatti (1999) has kindly provided us with his AIM
results of periodic RHF calculations using the geometry from
the 123 K SCM study and three different basis sets:
6-21G(d,p), 6-31G(d,p) and 6-311G(d,p). The discussions of
the present work refer to these new calculations (Gatti, 1999),
which are slightly different from the published ones (Gatti et
al., 1994; Gatti & Cargnoni, 1997).
The crystal structure of urea shows several chemically
remarkable features. Fig. 1 shows the molecular structure and
labelling scheme. The molecule adopts a ¯at conformation in
the crystal with site symmetry mm2. In contrast, the isolated
molecule is non-planar, the NH2 groups being twisted slightly
out of the molecular plane as determined by microwave
spectroscopy (Godfrey et al., 1997) and by ab initio calculations (Dixon & Matsuzawa, 1994; Ha & Puebla, 1994; Rasul et
al., 1994; Godfrey et al., 1997; Rousseau et al., 1998; Miao et al.,
2000). It is now accepted that the C2 molecular structure is the
most stable followed by a Cs conformation. The C2v structure
found in the crystal is a saddle point separating these two
stable conformers. Another most interesting aspect of the
crystal structure of urea is the hydrogen-bond pattern where
the O atom accepts four hydrogen bonds, see Fig. 1. This is
extremely rare. The only other such structures we are aware of
are found in the intriguing series of host±guest molecular
complexes based on a host derived from calix[4]pyrrole
(Bonomo et al., 1999; Anzenbacher et al., 1999).1
Urea is known to undergo several phase transitions at
moderate pressures (<0.4 GPa at room temperature). The
structure of the ®rst high-pressure phase has been solved
(Bonin et al., 1999) but the coordinates are not available at the
present time. However, it is clear that the columnar structure
(Fig. 1b) collapses. The molecule becomes non-planar as in the
gas phase and the O atom accepts only three hydrogen bonds.
Thus, small pressure is suf®cient to distort the open columnar
structure at the expense of a reduction of the number of
hydrogen bonds per O atom.
Here we report a high-precision study of the charge density
of urea using synchrotron diffraction data collected at the
Swiss±Norwegian Beam Lines (SNBL) at the European
Synchrotron Radiation Facility (ESRF), Grenoble, France.
The use of synchrotron radiation led to a data set of unprecedented resolution and precision, which in turn allowed the
use of a very extensive multipole model. The charge density
thus obtained can be used as a benchmark for testing theory
that, as will become apparent, is still lagging behind experiment in this case.
2. Experimental
All diffraction measurements were performed at the bending
magnet based Swiss±Norwegian Beam Lines (SNBL) at ESRF.
2.1. Powder diffraction ± thermal expansion and anisotropic
strain
The sample was mounted in a 0.7 mm borosilicate capillary.
High-resolution powder data were collected at the SNBL
Ê , was calibrated with
B-station. The wavelength, = 0.64791 A
an Si standard. Data were collected at four temperatures: 295,
120, 80 and 10 K. For the three low-temperature collections,
the temperature was controlled by a modi®ed STVP-400
open-¯ow helium cryostat from Janis Research Company. The
step size, 2, was 0.005 in all measurements. For 295, 120
and 80 K, the counting time was 1 s stepÿ1 while, for 10 K, it
was 4 s stepÿ1 for 2 = 9±19 and 8 s stepÿ1 elsewhere. The
data sets cover 8.5±21.995 , except for the 10 K data set which
starts at 9.0 . The powder patterns displayed signi®cant
anisotropic peak broadening. LeBail ®ts (LeBail et al., 1988)
were performed with GSAS (Larson & Von Dreele, 1994)
using the Stephens (1999) anisotropic broadening model in
combination with the Thompson et al. (1987) parameterization
of the pseudo-Voigt function and the Finger et al. (1994) peak
Figure 1
(a) Molecular structure of urea and labelling scheme; (b) perspective
view of the crystal structure along the a axis. The 50% probability
ellipsoids represent the harmonic ADPs at 123 K. Symmetry operation
(i) is x, 1 ÿ y, z.
372
Henrik Birkedal et al.
Charge density of urea
1
Ammonium cyanate was originally thought to provide another example of
fourfold acceptor O atoms, based on an X-ray powder structure (Dunitz et al.,
1998). However, it was recently shown by neutron powder diffraction that the
hydrogen-bond acceptor is actually not the cyanate O atom but rather the
cyanate N atom, which acts as a fourfold acceptor (MacLean et al., 2003).
Acta Cryst. (2004). A60, 371±381
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asymmetry correction. The ®ts with observations and a table
containing the pro®le parameters are given in the supplementary material (Fig. S1, Table S1), which also contains a
more detailed discussion of the anisotropic broadening.2
The resulting lattice parameters are shown in Fig. 2. They
increase smoothly with increasing temperature. The neutron
measurements are slightly off the curves, are somewhat larger
and show a much larger scatter. The thermal expansion is
evidently strongly anisotropic (Swaminathan, Craven &
McMullan, 1984). It was modelled by a single effective
Einstein oscillator, ln(V=V0) = (kBn=BVm){=[exp(=T) ÿ
1]}, where V0 is the cell volume at zero temperature, kB the
Boltzmann constant, n the number of atoms per unit cell, B
the bulk modulus, Vm the molar volume, a GruÈneisen
parameter and the effective Einstein temperature (Wang &
Reeber, 2000). Since we have not measured B, the constants
were combined into an effective pre-factor AE = kBn=BVm.
The ®ts have correlation coef®cients larger than 0.9996, the
Ê , (a) =
®nal ®tting parameters are V0(a) = 5.5640 (11) A
Ê , (c) =
280 (31) K, AE(a) = 9.4 (5), V0(c) = 4.68316 (6) A
268 (6) K and AE(c) = 2.19 (2). Interestingly, the Einstein
temperatures of a and c are essentially equal so that the
anisotropy of the thermal expansion stems from the prefactor, which includes the GruÈneisen parameter and the
compressibility. The difference in stiffness along the two
directions appears to be responsible for the difference in
thermal expansion.
The powder diffraction patterns display anisotropic microstrain broadening and the peak widths increase with
decreasing temperature.2 Tentatively, we suggest that this is
linked to a phase transition. No anomalies have been detected
in the heat capacity at ambient pressure (Ruehrwein &
Huffman, 1946; Andersson et al., 1993) but the transition
pressure of the ®rst pressure-induced phase transition is
reduced at lower temperatures (Andersson & Ross, 1994) and
it has been predicted that the high-pressure phase might
become thermodynamically stable at low temperatures. The
powder diagrams display neither peak splitting nor the
appearance of new peaks. However, Andersson & Ross (1994)
found that the phase transition could be induced only under
isothermal pressurization and not under isobaric cooling.
Hence a suppressed phase transition may be the reason for the
anisotropic strain broadening. Whether this phase transition is
suppressed by thermodynamics or by kinetics is unclear.
cally constructed for off-vertical plane diffractometry and this
use of the diffractometer has been discussed previously
(Thorkildsen et al., 1999, 2000). Here, we used the diffractometer as a standard four-circle instrument with the
detector circle (2) in the vertical plane to pro®t from the
linear polarization of the radiation.
Station A of SNBL offers two types of optics: parallel-beam
and focusing optics. In the parallel-beam con®guration, the
monochromator contains two ¯at Si(111) crystals arranged in
a ®xed-exit geometry. In the focusing set-up, the optics consists
of a vertically collimating Rh-coated Si mirror, a ®xed-exit
Si(111) monochromator where the second crystal is sagitally
bent for horizontal focusing, followed by an Rh-coated Si
mirror that provides vertical focusing. We collected test data
using both optical con®gurations. The best results, in terms of
internal consistency of the data sets, were obtained with
parallel-beam optics. We attribute this to somewhat more
stable beam conditions for this mode. For charge-density
studies, where the crystals are of `normal' size, the intensity
gained by focusing is not needed. Indeed, the main advantages
of synchrotron radiation are the low inherent background
(stemming from the low divergence) and the high degree of
monochromaticity, which make weak re¯ections much easier
to measure. Therefore, parallel-beam optics was chosen for
the ®nal data collection.
2.2.2. Charge-density data collection. An overview of the
single-crystal data-collection parameters is given in Table 1,
which compares our measurements with those of SCSS and
ZSTVF. The wavelength was determined from the roomtemperature lattice parameter of a CaF2 reference crystal
re®ned from the setting angles of 24 re¯ections. The datacollection temperature, controlled with an Oxford cryosystem
N2 cooling system, was 123 K to allow use of the SCM neutron
H-atom parameters in the charge-density re®nements. The cell
parameters were determined from the angles of 17 centred
re¯ections. The results are within one s.u. of the values
obtained from the powder diffraction data (Fig. 2). This very
2.2. Single-crystal measurements
2.2.1. Beamline layout and diffractometer. A signi®cant
effort was invested in the optimization of the data-collection
parameters. Station A of SNBL is equipped with a six-circle
diffractometer from KUMA diffraction (KM6-CH) speci®2
Supplementary material, including a CIF ®le and structure factors, ®gures
showing the quality of the Le Bail ®ts, tabulations of results of the Le Bail ®ts,
a discussion of the anisotropic peak broadening, tables and discussion of
topological properties of the electron density, and an overview of previous
experimental determinations of the dipole moment of urea in solution and gas
phase are available from the IUCr electronic archives (Reference: XC5013).
Services for accessing these data are described at the back of the journal.
Acta Cryst. (2004). A60, 371±381
Figure 2
Temperature dependence of the lattice parameters of urea. The solid lines
are ®ts to the lattice parameters from powder diffraction. Also shown are
the SCM (Swaminathan, Craven & McMullan, 1984) neutron and the
single-crystal (SC) parameters of the present work.
Henrik Birkedal et al.
Charge density of urea
373
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Table 1
Data collections on urea.
SCSS is Swaminathan, Craven, Spackman & Stewart (1984), ZSTVF is
Zavodnik et al. (1999). The two values under ZSTVF in the re®nement part of
the table
to the full and low-angle data set, respectively, see text.
P refer P
R ˆ …Fo2 †= Fo2 , calculated from the deposited data of SCSS and
ZSTVF. Rint for SCSS refers only to averaging of reference re¯ections.
Experimental
T (K)
Ê)
a (A
Ê)
c (A
Ê)
(A
(mmÿ1)
Crystal size (mm)
Absorption correction
Scan type
No. of re¯ections
measured
unique
Rint
R
hFo2 =…Fo2 †i
Ê ÿ1)
sin =max (A
TDS correction
Re®nement
Re®nement on
No. of parameters
wR(F)
R(F)
wR(F2)
R(F2)
S
Present
SCSS
ZSTVF
123 (2)
5.5780 (6)
4.6860 (7)
0.5996 (1)
0.07
0.090.130.23
None
!
123 (2)
5.578 (1)
4.686 (1)
0.7107
0.11
0.400.400.25
None
!±2
148 (1)
5.5890 (5)
4.6947 (4)
0.7069
0.127
0.300.300.35
Numerical
!±2
3942
1045
0.0141
0.0153
49.9
1.44
None
`A sector'
318
0.03
0.0242
29.3
1.15
None
1971
412/145
0.015
0.0045
148.5
1.15
Peak tail ®ts
F2
77
0.0054
0.0204
0.0108
0.0071
1.1661
F
34
0.015
0.035
F
28/29
0.028/0.011
0.025/0.008
1.57
1.64/1.09
good agreement indicates that the temperature of the singlecrystal experiment was estimated correctly.
The data collection took about three days and covered a
hemisphere of reciprocal space, selected so as to minimize
vibrations induced by the cold stream. Initial test measurements showed marked improvement when the sample mount
was kept roughly parallel to the gas stream. The crystals had a
tendency to crack when the temperature was lowered. Neither
of the previous works mention this problem. It is possible that
the cracking only becomes visible at the increased resolution
of the synchrotron. Cooling was most successful with a sample
that had a somewhat broad room-temperature mosaic spread
because a slightly strained crystal probably gives better data
than a cracked one. Re¯ection intensities were measured with
! scans that, owing to the negligible dispersion of the
synchrotron beam, can be expected to give the best peak
sampling (Blessing, 1987). The scan width was kept constant at
! = 1.0 . A small set of strong low-angle re¯ections was
measured with a scan width of 1.2 to verify that scan-truncation errors did not occur. The re¯ections were scanned
continuously using 100 bins per re¯ection. This is faster than
and at least as reliable as a step scan with 100 steps. The
strongest re¯ections were measured with Cu foil attenuators.
For most of them, the standard pre-detector attenuator was
suf®cient, but for the very strongest re¯ections additional
attenuators were placed in the incident beam. Attenuator
absorptions were derived from measurements with and
without ®lter.
374
Henrik Birkedal et al.
Charge density of urea
Ê ÿ1 (Table 1), while the sample
The data set extends to 1.44 A
scattered to higher angles. However, at the highest angles,
2 ' 120 , the detector arm was at risk of colliding with other
equipment in the experimental hutch.
2.2.3. Data reduction. Data reduction was performed using
the XD_RED program of Mathiesen (2001), updated to
accommodate synchrotron data from the KM6-CH diffractometer. It performs detailed error propagation calculations
with inclusion of, in particular, the s.u.'s of the degree of
polarization and the wavelength. The background was estimated from the ®rst 9 and the last 9 steps of each scan. The
degree of polarization of the polychromatic synchrotron beam
was set to 0.95 (1). This value has been measured in earlier
experiments at SNBL under slightly different conditions
(Birkedal, 2000). XD_RED takes the effect of the monochromator explicitly into account and obtained a degree of
polarization at the sample position of 0.952. The variance of a
re¯ection includes an instrumental instability factor of s =
0.002, according to 2 …FH2 † ˆ 2 ‰IH …net† ‡ ‰sIH …net†Š2, where
IH is the intensity.
2.2.4. Thermal diffuse scattering. Thermal diffuse scattering (TDS) may be large at high diffraction angles and at
temperatures signi®cantly above the Debye temperature, D .
D of urea has been derived from heat-capacity measurements (Andersson et al., 1993), using several different
methods. The smallest value obtained was D = 135.0 K. The
present experiment, as well as that of SCSS, was performed at
123 K, below the Debye temperature. It was thus deemed safe
to neglect the TDS correction, considering also that the
temperature dependence of the elastic constants is unknown.
ZSTVF corrected for TDS at 148 K by making linear ®ts to the
tails of the diffraction peaks. In view of the anisotropic peak
broadening, this procedure may lead to questionable results
for our crystal.
2.2.5. Charge-density refinement. The VALRAY program
system (Stewart et al., 1998) was used for the full-matrix leastsquares (LSQ) re®nement based on F2 and using experimental
weights, w…H† ˆ 1= 2 ‰Fo2 …H†Š. All re¯ections were included in
the re®nement. The Hamilton (1965) R-factor-ratio test was
used to verify that newly introduced parameters led to
signi®cant improvement in residuals.
The initial fractional coordinates and ADPs of N, C and O
atoms were taken from SCSS. H-atom nuclear parameters
were taken from the 123 K SCM neutron study and kept ®xed
during the re®nement. Anharmonicity was included via
re®nement of third-order Gram±Charlier coef®cients on N
and O atoms, which led to signi®cant improvement in residuals. Tests employing third-order parameters on C atoms as
well did not lead to lower residuals. The multipole expansion
was extended up to the hexadecapole level for C, N and O
atoms and to the quadrupole level for H atoms. At all levels,
population parameters above 3 s.u. were found
The radial functions of mono-, di- and quadrupoles for C, N
and O atoms were the density-localized (van der Wal &
Stewart, 1984) orbital products of Clementi & Roetti (1974).
Let (is) designate an inner shell density localized orbital
product and (os) an outer shell product. The following
Acta Cryst. (2004). A60, 371±381
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Table 2
Parameters varied in the charge-density re®nement.
The neutron results used for H atoms are from SCM. Radial functions are:
² density localized orbital products, * re®ned; ³ exponential, re®ned (SCSS
obtained from multipole ®ts to isolated molecule RHF/6-31G(d,p) structure
factors); exponential, re®ned. IAM = independent atom model.
Position
Uij
Cijk
H atoms
Poles:
mono
di
quadru
octu
hexadeca
Extinction
Present
SCSS
ZSTVF
C, N, O
C, N, O
N, O
Neutron
Neutron
Neutron
High-angle IAM
High-angle IAM
²
²
Neutron
²
³
C,N,O,H
C², N², O², H³
C², N², O², H³
C³, N³, O³
C³, N³, O³
Becker & Coppens
(1974)
³
³
C,N,
C³, N³,
C³, N³,
C³, N³,
None
³
³
O,H
O³, H³
O³, H³
O³
Scaled neutron
2.3. Analysis of the electron density: atoms in molecules
C*, N*, O*, H*
C*, N*, O*, H*
C*, N*, O*
C*, N*, O*
C*
Zachariasen (1967)
The electron density was analysed in terms of the theory of
atoms in molecules (Bader, 1994). Integrated properties were
obtained using the method of Flensburg & Madsen (2000). A
similar algorithm has been used by Gatti (1999). For the
integration of the experimental density, 5024 points were used
to span the surface of C and N atoms while 2788 points were
determined on the zero-¯ux surface of O atoms. For H atoms,
664 points were employed for each zero-¯ux surface. This gave
an integration sampling between 105 and 106 points in each
atomic basin.
products were used: core monopoles (is)(is), valence monopoles (os)(os), valence dipoles (2p)(os), and valence quadrupoles (2p)(2p). For the valence mono-, di- and quadrupoles, a
common parameter was re®ned for each atom. The radial
functions of the higher poles (octu- and hexadecapoles)
on C, N and O atoms and all radial functions on H atoms
were described by exponential functions, Rl …r† =
‰nl ‡3 =…nl ‡ 2†!Šrnl exp…ÿl r†, with nl l (Stewart, 1977). The
initial values of were standard molecular values (Hehre et
al., 1969, 1970) and nl was set equal to l. In the ®nal model, the
's were re®ned. The parameters of the octu- and hexadecapoles were constrained to be equal for each atom type
while all 's of the two H atoms were constrained to be equal.
The core monopole populations were constrained to be equal,
Pc(C) = Pc(O) = Pc(N).
In Table 2, the present re®nement is compared with the
re®nements of SCSS and ZSTVF. Our ®nal model comprised
77 parameters. In addition to the nuclear and electron parameters shown in the table, an isotropic extinction parameter,
g, with Lorentzian distribution of mosaic blocks (Becker &
Figure 3
Residual charge density in urea after multipole re®nement, view on the
molecular plane. Positive contours full lines, negative contours dashed,
Ê ÿ3. Map on the left calculated
zero contour long dashes, intervals 0.05 e A
Ê ÿ3. Map
with all re¯ections, maximum and minimum contours 0.20 e A
on the right calculated with the resolution of SCSS and ZSTVF,
Ê ÿ1.
sin =max = 1.15 A
Acta Cryst. (2004). A60, 371±381
Coppens, 1974) was re®ned. The ®nal value g =
0.016 (3) 10ÿ4 rad2 yields a lowest Fo2 =Fc2 ratio of 0.974 for
the 110 re¯ection ± well within the regime where the
description of extinction is adequate. The scale factor was not
re®ned but estimated from the ratio between F(000) and the
sum of valence and core populations (Stewart, 1976). Its ®nal
value is 0.952 (6). The largest correlation coef®cient was 0.95
between C and P30(C).
3. Discussion of the single-crystal study
The internal agreement of our data set is comparable to the
two previous studies, SCSS and ZSTVF, even though both the
resolution and the number of averaged re¯ections are much
larger. In Table 1, we compare the three data sets,3 and in
particular the values of R and hFo2 =…Fo2 †i whose signi®cance
depends of course on a proper estimation of the s.u.'s. The
present values are signi®cantly superior to those of SCSS as
should be expected from the combination of repeated
measurements and the bene®ts of synchrotron radiation. The
values given by ZSTVF are even more impressive. For the
fraction of our data set with the resolution of ZSTVF, we
obtain R = 0.0067 and hFo2 =…Fo2 †i = 80.5. A more detailed
analysis of hFo2 =…Fo2 †i for different groups of re¯ections and
the rather large value of the goodness of ®t S of ZSTVF
suggests that ZSTVF may have underestimated the s.u.'s. In
addition, the ZSTVF data set was corrected for TDS by ®tting
peak tails, but the anisotropic strain broadening observed in
our powder measurements suggests other contributions to the
tails.
Our re®nement led to very small R values (Table 1)
demonstrating the extremely good agreement between
measurements and model. The residual density is shown in
Fig. 3, which displays two maps; one map has been calculated
with all re¯ections and the other one with the resolution
Ê ÿ1 of the two previous studies. The latter is
sin =max = 1.15 A
completely ¯at while the former shows features amounting to
Ê ÿ3. This is due to the expected increase of the
0.20 e A
relative uncertainty …Fo2 †=Fo2 at higher angles. It can thus be
concluded that the re®nement successfully reproduces the
measured intensities.
3
For these comparisons, the data sets of SCSS and ZSTVF were transformed
from F to F2. The s.u.'s were transformed according to (F2) = 2F(F).
Henrik Birkedal et al.
Charge density of urea
375
research papers
Table 3
Table 4
First line: present study; second line: 123 K neutron results of SCM. H-atom
parameters are not included as they were taken from the neutron re®nement.
Ê 2, Cijk
The fractional coordinates are multiplied by 104, Uij in units of 10ÿ4 A
multiplied by 105. C and O atoms are on positions (0, 1/2, z), site symmetry
2.mm, the N atom is on position (x, x+1/2, z), site symmetry ..m.
HSP are the standard molecular values used as starting point in the re®nement
(Hehre et al., 1969, 1970).
Ê ÿ1) for urea.
Radial scaling parameters (A
Nuclear parameters of urea.
x
z
U11
U33
U12
U13
1446.8 (7)
1447 (1)
3282.1 (3)
3280 (2)
5963.4 (10)
5962 (2)
1790.1 (10)
1785 (1)
151.9 (4)
147 (5)
195.8 (4)
197 (6)
293.1 (6)
286 (4)
67.5 (4)
65 (3)
66.5 (3)
63 (4)
95.6 (4)
95 (2)
ÿ3.5 (6)
1 (4)
16.4 (8)
17 (5)
ÿ157.0 (7)
ÿ147 (2)
0.2 (3)
2 (3)
C123
C133
C333
2 (4)
ÿ36 (3)
1.2 (7)
1.3 (5)
1.8 (6)
C
O
N
O
N
C111
C112
ÿ6.5 (7)
5.8 (8)
C113
ÿ3.2 (9)
18.8 (13)
3.1. Geometry and thermal motion
Fractional coordinates and displacement parameters are
given in Table 3. The agreement between the nuclear
parameters obtained in the present study and the neutron
results of SCM is excellent. All quantities are equal within two
neutron s.u.'s except for z and U12 of the N atom where the
difference is 5.1 and 5.0 neutron s.u.'s, respectively. This
difference is due to the inclusion of anharmonicity in the
present model, with particularly signi®cant results for the N
atom.
The SCM neutron displacement parameters were analysed
by Capelli et al. (2000). They adjusted by LSQ a molecular
mean-®eld model of rigid-body and low-frequency internal
motion to the temperature dependence of the ADPs (BuÈrgi &
FoÈrtsch, 1999; BuÈrgi & Capelli, 2000; BuÈrgi et al., 2000). They
found that the pyramidalization of the N atom, which is partly
responsible for the non-planarity of the molecule in the gas
phase, is also important in the solid state. The largest anharmonic effect in the present model is in the out-of-plane
N-atom motion, consistent with the analysis of BuÈrgi et al.
(2000). The molecular geometry and the geometry of the
hydrogen bonds4 are given in Table S4 (supplementary
material). The geometry agrees with the neutron data but the
present results are signi®cantly more precise. The molecular
geometry in the gas phase (Godfrey et al., 1997) presents some
interesting differences from the solid state. The C O bond is
Ê shorter in the gas phase while the CÐN bond is
0.036 A
Ê longer. This difference is larger than what can be
0.040 A
expected from the non-planarity only and has been explained
as a direct effect of the extensive hydrogen bonding in the
crystal (Keuleers et al., 1999).
Present
C
O
N
H
SCSS
ZSTVF
HSP
0.991 (4)
0.9784 (16)
1.002 (3)
5.09 (10)
9.1 (6)
7.2 (2)
4.38 (5)
4.42 (8)
9.6 (3)
7.97 (19)
4.91 (17)
0.97 (1)
0.98 (1)
0.98 (1)
1.02 (2)
6.50
8.50
7.37
4.69
3.2. Deformation density model
The re®ned multipole population parameters are presented
in Table S3 (supplementary material). The present model
extends further than the two previous ones since hexadecapoles are used on all heavy atoms, see Table 2. ZSTVF used
hexadecapoles on C atoms only with the largest population
parameter being only 3 s.u. In contrast, our hexadecapole
parameters attain values of up to 6 s.u.'s. The current radial
description is more ¯exible than either of the previous studies
(Table 2). The exponents of the single exponentials used by
SCSS, one for each atom type, were not experimentally
derived but calculated from a LSQ ®t of an electron-density
model to theoretical structure factors obtained by a
6-31G(d,p) RHF molecular calculation assuming a crystal
consisting of non-interacting molecules. ZSTVF performed
re®nements but do not specify the radial functions used.
The re®ned radial parameters are given in Table 4, where
we compare them with the SCSS, ZSTVF and the standard
molecular exponents of Hehre et al. (1969, 1970), hereafter
referred to as HSP. Fig. 4 shows the radial function for the
C-atom octupole. The lower-order poles (l 2), treated by
re®nement, were only slightly modi®ed in the re®nement. C
and O atoms are expanded ( < 1), but C atoms only slightly
so. The inner poles on the N atom are unchanged within less
than one s.u. For the higher poles, the O atom is contracted
with respect to the HSP standard values while C, N and H
atoms are more diffuse. In comparison with SCSS, where the
values were re®ned with a scaling parameter common for all
poles, the C atom is less diffuse while the other atoms are more
4
All geometrical quantities involving H-atom positions and/or the positions of
critical points have been calculated from the parameters after re®nement. For
quantities involving H atoms, the s.u.'s given by SCM were used. This is
necessary as neither H-atom nor critical-point coordinates are variables in the
least-squares-re®nement process. The procedure neglects correlation between
the parameters.
376
Henrik Birkedal et al.
Charge density of urea
Figure 4
Normalized C-atom octupole radial function of urea.
Acta Cryst. (2004). A60, 371±381
research papers
diffuse. The C atom shows the largest variations. The
maximum of the radial function (Fig. 4) is rmax = nlÿ1, which
Ê for the HSP standard exponent, the
gives 0.46, 0.59 and 0.68 A
present re®ned one and the SCSS value, respectively. It is clear
that the C atom in urea has more diffuse components than the
standard HSP atom.
The need for ¯exible radial functions has been noted by
Figgis et al. (1993) in their 9 K X±N study of (ND4)2(Cu(SO4))2 6D2O. They conclude `The molecular density is
not described by simple modi®cation of atomic radial
functions'. They solved the problem by adding extra diffuse
radial functions on some atoms but did not re®ne exponents.
Volkov et al. (2000) investigated the difference between
experimental and theoretical crystal charge densities by
applying a multipole model to calculated structure factors.
Theoretical structure factors at the periodic RHF/6-21G(d,p)
and RHF/6-31G(d,p) level were calculated to a resolution of
Ê ÿ1. The ensuing multipole re®nements were
sin = < 1.05 A
subjected to local symmetry and chemical constraints and
extended to octupoles only for the non-H atoms and to the
quadrupole level for H atoms. Their principal conclusion was
`The main origin of the observed discrepancies is attributed to
the nature of the radial functions in the experimental multipole model'. While their multipole modelling approach can be
considered somewhat restricted, this conclusion shows that it
is necessary to allow full ¯exibility in the radial functions.
3.3. Critical points
In Tables 5, S5, S6, S7 and S8, we present the critical points
(CPs) of the electron density and compare them with the
theoretical and ZSTVF results. The topology ful®ls the Morse
equation, n(3, ÿ3) ÿ n(3, ÿ1) + n(3, +1) ÿ n(3, +3) = 0 for
periodic systems, and is thus consistent. A qualitative discussion of the topology is given in the supplementary material.
Figure 5
Relation between bond length D and (rc) for CÐO bonds. Systems are
citrinin (Roversi et al., 1996), MAHS (methylammonium hydrogen
succinate, Flensburg et al., 1995), MADMA (methylammonium hydrogen
maleate, Madsen et al., 1998a), benzoylacetone (Madsen et al., 1998b),
d,l-Asp (Flaig et al., 1998), Gly-l-Thr dihydrate (Dittrich et al., 2000) and
urea from the present study. Two exponential ®ts are given, one to the
citrinin data `Roversi et al. (1996)' and one to all the data `new ®t'.
Acta Cryst. (2004). A60, 371±381
The hydrogen-bonded columns create empty tunnels along the
c axis (Swaminathan, Craven & McMullan, 1984). In these, the
atomic surfaces of the N atoms meet, and the N atoms are thus
bonded (in the topological sense) across the tunnel. The
topology derived from the theoretical calculations (Gatti,
1999) shows exactly the same set of bond CPs (BCP).
The only qualitative difference between the ab initio and
experimental electron densities is in the regions where the
electron density is close to zero. In the experimental
density, there is a minimum (C5) with negative (rc) =
Ê ÿ3, i.e. practically zero. In addition, there
ÿ0.0008 (2) e A
is a minimum with a positive density (C4) with (rc) =
Ê ÿ3. The ®rst is situated in the `walls' of the
0.0030 (1) e A
empty column mentioned above while the latter is situated
close to the centre. Two ring critical points are also present
(R5 and R6, Table S6) to ful®l the Morse equation. In the
theoretical density, R5 and R6 are substituted by a single point
R50 (see Table S7 for characteristics of the ring and cage
critical points of the periodic ab initio calculations). The
positivity of experimental electron densities is not guaranteed
by the multipole model, while the ab initio calculations do not
suffer from this problem. The local de®ciency of the experimental model is not expected to have major consequences
outside low-density regions.
Roversi et al. (1996) found that the value of the charge
density in the BCP of CÐO bonds in citrinin follows the
Ê
relation (rc) = 6.063Dÿ3.521, where the bond length D is in A
ÿ3
Ê
and (rc) in e A . We included several other values from the
literature to get an idea of the quality of the relation, Fig. 5.
The agreement is generally very good and the present value
for urea is in extremely good agreement with the literature
values. By ®tting the curve to all the values, a new relation is
obtained which is labelled `new ®t' in Fig. 5:5 (rc) =
6.087(180)Rÿ3.498(120). Our value of (rc)(CÐO) is larger by
Ê ÿ3 than that of ZSTVF. The agreement of our value
0.186 e A
with the host of other experimental values (Fig. 5) suggests
that the result of ZSTVF is less accurate. The present value of
the CÐO Laplacian is about the same as has been found in the
studies presented in Fig. 5. It is also close to the average values
derived from nine amino acids by Flaig et al. (1999), who
Ê ÿ5 for the long and short
found ÿ33.0 (40) and ÿ34.9 (26) e A
carboxylate bond, respectively. The value of ZSTVF,
Ê ÿ5, seems to be too small.
ÿ18.86 e A
The bond length and (rc) places the CÐO bond between a
typical double and single bond. The value of the ellipticity
indicates a predominantly -type bond. The CÐN bond is
rather short, and shows relatively large values of (rc) and of
ellipticity. This indicates a contribution to the bond with
charge delocalization over the O. . .C. . .N framework. The
intermolecular interactions are all of closed-shell nature
(Table S5), in agreement with the expectation for normal
predominantly electrostatic hydrogen bonds.
5
No s.u.'s on (rc) are given by Flaig et al. (1998) for d,l-Asp. For the LSQ ®t,
the four observations from this study were, arbitrarily, assigned s.u.'s of 0.04 in
order to permit using weighted LSQ.
Henrik Birkedal et al.
Charge density of urea
377
research papers
Table 5
Intramolecular bond critical points in crystalline urea.
D is the bond length AÐB, D1 is the distance from A to the BCP while D2 is the B±BCP distance. First line: present study; second: periodic RHF 6-21G(d,p); third:
periodic RHF 6-31G(d,p); fourth: periodic RHF 6-311G(d,p); ®fth: experimental density of ZSTVF.
CÐO
CÐN
NÐH1
NÐH2
Ê)
D (A
Ê)
D1 (A
Ê)
D2 (A
Ê ÿ3)
(rc) (e A
Ê ÿ5)
r2(rc) (e A
Ê ÿ5)
1 (e A
Ê ÿ5)
2 (e A
Ê ÿ5)
3 (e A
"
1.2565 (5)
1.258
1.258
1.258
1.258
1.3384 (4)
1.341
1.341
1.341
1.343
1.005 (2)
1.007
1.007
1.007
1.005
1.0020 (15)
1.000
1.000
1.000
0.997
0.461 (3)
0.430
0.410
0.421
0.427
0.542 (4)
0.471
0.449
0.466
0.440
0.751 8(12)
0.763
0.785
0.769
0.795
0.7486 (18)
0.755
0.778
0.762
0.791
0.796 (3)
0.827
0.848
0.837
2.722 (10)
2.545
2.594
2.578
2.536
2.359 (16)
2.326
2.376
2.339
2.538
2.25 (2)
2.360
2.341
2.311
1.797
2.280 (13)
2.405
2.384
2.355
1.843
ÿ31.8 (3)
ÿ12.90
ÿ7.46
ÿ11.92
ÿ18.86
ÿ25.0 (3)
ÿ22.86
ÿ27.35
ÿ26.53
ÿ37.66
ÿ30.9 (7)
ÿ48.78
ÿ47.27
ÿ48.54
ÿ30.44
ÿ34.6 (3)
ÿ49.67
ÿ48.20
ÿ49.45
ÿ33.37
ÿ25.08 (15)
ÿ24.40
ÿ25.37
ÿ24.62
ÿ23.33
ÿ20.29 (17)
ÿ21.77
ÿ21.51
ÿ20.19
ÿ27.34
ÿ31.2 (3)
ÿ35.65
ÿ35.10
ÿ33.53
ÿ26.77
ÿ32.31 (19)
ÿ36.24
ÿ35.66
ÿ34.10
ÿ28.13
ÿ24.38 (16)
ÿ23.50
ÿ25.29
ÿ24.31
ÿ20.23
ÿ16.2 (2)
ÿ19.09
ÿ19.70
ÿ18.55
ÿ23.00
ÿ28.6 (4)
ÿ33.92
ÿ33.50
ÿ31.90
ÿ25.54
ÿ30.10 (16)
ÿ34.45
ÿ34.02
ÿ32.43
ÿ26.93
17.66 (12)
34.99
43.20
37.01
24.71
11.50 (16)
18.00
13.86
12.20
12.68
28.94 (13)
20.79
21.33
16.89
21.88
27.78 (12)
21.01
21.48
17.07
21.68
0.029 (9)
0.038
0.003
0.013
0.15
0.26 (2)
0.140
0.092
0.089
0.19
0.091 (19)
0.051
0.048
0.051
0.05
0.073 (9)
0.052
0.048
0.052
0.05
0.796 (4)
0.870
0.892
0.874
0.253 (2)
0.244
0.222
0.238
0.253 (2)
0.245
0.222
0.239
3.3.1. Quantitative comparison of BCPs with periodic RHF
and ZSTVF. Tables 5 and S5 compare the BCPs with the
periodic RHF results and the data of ZSTVF. The CÐO and
CÐN BCPs are always closer to C than to O and N atoms, and
in the NÐHi bonds they are closer to the H than to the N
atom. It is clear from the distances that both the periodic RHF
calculations and ZSTVF show systematically more polarized
bonds than the present density. Compared to our experimental
density, (rc) of the CÐO and CÐN bonds is underestimated
by the RHF calculations while those in NÐH bonds are
overestimated. In the ZSTVF study, the CÐO and NÐH
bonds show smaller, and the CÐN bond larger, electron
densities than in the present.
The Laplacians also show large variations. Compared to the
present results, the CÐO Laplacian is consistently too small in
the RHF calculations while the NÐH values are too large. For
the CÐO bond, the erratic behaviour upon extension of the
basis set indicates that the basis sets are inadequate. For all
three basis sets, and especially for 6-31G(d,p), 3 is predicted
to be far too large. For the NÐH bonds, 1 and 2 are overestimated (too negative) while 3 is too small. As mentioned in
the previous section, ZSTVF gives too small a Laplacian for
the CÐO bond. This is largely due to too large a value of 3.
In order to investigate the effect of basis set and electron
correlation on CPs and integrated properties, we performed a
large number of computations on urea (Birkedal & Schwarzenbach, 2004). These included non-correlated calculations
(RHF), perturbation treatment of correlation (MP2, MP3,
MP4SDQ), variational treatment of correlation (CISD) and a
single density functional approach (B3LYP). Computations
were done with different basis sets: 6-31G(nd,np) (n = 1, 2, 3),
6-311G(nd,np) (n = 1, 2, 3), 6-311++G, 6-311++G(3d,3p), ccpVDZ and cc-pVTZ. These include some of the most
frequently used double- and triple-zeta basis sets. We shall not
378
Henrik Birkedal et al.
Charge density of urea
discuss these results in detail here but merely summarize some
key results. For both the 6-31G(nd,np) and 6-311G(nd,np)
series, it was found that the above-mentioned over-polarization of the CÐO bond disappeared when more polarization
functions were added [increasing n in (nd,np)]. This resulted in
a larger absolute (more negative) value of the Laplacian at the
bond critical point. The addition of diffuse functions did not
in¯uence this property. This shows that the oft-used basis sets
6-31G(d,p) and 6-311G(d,p), with or without diffuse functions,
are not well adapted for the description of the strongly
polarized CÐO bond. The agreement with the present study
was further improved by adding electron correlation.
These observations consistently indicate a de®ciency in the
computations, especially for the highly polarized CÐO bond.
The ZSTVF density shows many of the same problems as the
calculations.
3.3.2. Integrated properties. Integrated atomic Laplacians,
charges, volumes and dipole moments are given in Table 6,
where they are compared with the corresponding values
derived directly from the multipole model. This table also
reports the properties obtained by integration of the theoretical data (Gatti, 1999). The superior precision of the present
study is immediately seen by comparison of the monopole
charges with those of ZSTVF, the monopole s.u.'s being lower
by a factor of 2±3.5 in the present work. The ZSTVF monopole charges are equal to the present ones within three
ZSTVF s.u.'s.
The integrated Laplacians, L, are all reasonably small
indicating that the integrations over the atomic basins are
suf®ciently precise. This is re¯ected by the integrated volumes
and charges that all deviate by less than 0.3% from the
expected values (Vmolecule = 1=2Vcell and qmolecule = 0).
The integrated atomic charges obtained from the periodic
RHF calculations are much larger than the experimental ones,
Acta Cryst. (2004). A60, 371±381
research papers
Table 6
Integrated properties for urea: Laplacian, charges and dipole moments.
Ê ÿ2. Integrated volume V in A
Ê 3 units, the three charges (q, qmono and qmono
L is the integrated Laplacian in units of 10ÿ5 e A
ZSTVF ) in e units. The right half of the table
Ê . The only non-zero dax and day are dax …N† = 0.023, dax …H1† = 0.055 and
gives the atomic dipole moments and their contributions. All dipole contributions in units of e A
Ê . The ®rst line gives the results of the present study, the three following lines present the periodic RHF results (Gatti, 1999) in the order
day …H1† = ÿ0.002 e A
P
P
P
6-21G(d,p) (second line), 6-31G(d,p) (third line) and 6-311G(d,p) (fourth line). NH2, CO, mol give sums over the NH2 group, the CO group and the full
molecule, respectively.
Dipole moments
Charges
AIM
Multipoles
L
V
q
qmono
qmono
ZSTVF
daz
dctz
dz
daz
dctz
dz
C
19
ÿ1176
ÿ1192
ÿ701
20
49
31
9
478
83
447
190
ÿ484
ÿ29
37
12
ÿ251
ÿ3
5
48
4.35
3.33
3.07
3.34
17.58
17.87
18.51
18.52
19.52
19.78
20.19
19.72
2.91
2.97
2.65
2.85
2.98
3.13
2.81
2.98
25.40
25.88
25.64
25.55
21.93
21.20
21.59
21.85
72.73
72.95
72.87
72.96
1.667
2.322
2.478
2.332
ÿ1.177
ÿ1.429
ÿ1.479
ÿ1.431
ÿ1.214
ÿ1.442
-1.572
ÿ1.453
0.482
0.503
0.541
0.507
0.493
0.492
0.531
0.497
ÿ0.240
ÿ0.448
ÿ0.500
ÿ0.450
0.490
0.893
0.999
0.900
0.011
ÿ0.002
ÿ0.000
ÿ0.000
ÿ0.07 (4)
ÿ0.03 (13)
C
ÿ0.12 (6)
O
ÿ0.071 (7)
ÿ1.13 (6)
ÿ1.21 (6)
ÿ0.04 (5)
ÿ0.30 (9)
N
0.016 (8)
ÿ0.03 (4)
ÿ0.02 (4)
0.14 (2)
0.21 (4)
2.677
3.620
3.860
3.639
ÿ3.146
ÿ3.773
ÿ3.871
ÿ3.791
ÿ1.048
ÿ1.263
ÿ1.380
ÿ1.276
0.615
0.638
0.686
0.641
ÿ0.016
ÿ0.013
ÿ0.014
ÿ0.012
ÿ0.448
ÿ0.639
ÿ0.708
ÿ0.648
ÿ0.469
ÿ0.153
ÿ0.012
ÿ0.153
ÿ1.365
ÿ1.430
ÿ1.427
ÿ1.449
ÿ0.11 (6)
ÿ0.23 (7)
2.564
3.571
3.812
3.586
ÿ3.290
ÿ3.995
ÿ4.136
ÿ4.002
ÿ1.018
ÿ1.194
ÿ1.302
ÿ1.204
0.641
0.670
0.721
0.675
ÿ0.080
ÿ0.080
ÿ0.087
ÿ0.081
ÿ0.458
ÿ0.604
ÿ0.667
ÿ0.610
ÿ0.725
ÿ0.424
ÿ0.324
ÿ0.415
ÿ1.640
ÿ1.632
ÿ1.659
ÿ1.635
ÿ0.017 (9)
ÿ0.41 (2)
0.113
0.049
0.048
0.052
0.144
0.221
0.265
0.210
ÿ0.029
ÿ0.069
ÿ0.078
ÿ0.073
ÿ0.026
ÿ0.032
ÿ0.035
ÿ0.034
0.064
0.067
0.072
0.069
0.009
ÿ0.034
ÿ0.040
ÿ0.038
0.257
0.270
0.312
0.263
0.275
0.202
0.232
0.186
0.088 (9)
0.19 (3)
0.28 (3)
ÿ0.222 (13)
ÿ0.022 (3)
ÿ0.244 (14)
ÿ0.119 (18)
0.13 (5)
0.01 (5)
ÿ0.088 (11)
ÿ1.24 (9)
ÿ1.33 (9)
ÿ0.33 (3)
ÿ0.98 (11)
ÿ1.30 (11)
O
N
H1
H2
P
NH2
P
CO
P
mol
H1
H2
P
NH2
0.24 (6)
0.12 (11)
ÿ0.48 (5)
ÿ0.26 (15)
P
CO
0.00 (9)
ÿ0.02 (21)
P
mol
especially for C atoms, where all three RHF calculations give
charges larger than 2.3 e. The aforementioned investigation of
basis set and correlation dependence of the BCPs was
extended to integrated atomic properties (Birkedal &
Schwarzenbach, 2004). The conclusion was clear: the very
large atomic charge on the C atom is due to the neglect of
electron correlation. The present experiment provides a target
value against which new calculations can be validated.
The molecular dipole moment P
vector can be written
as a
P
sum over atomic contributions d = q(
)X
+ M(
) =
dct + da, where X
is the nuclear position vector and M(
) is
the R®rst moment of the atomic charge distribution: M(
) =
ÿe r
(r) dr (Bader, 1994). For the following, we de®ne a
Cartesian coordinate system with the z axis along the CÐO
bond and the y axis perpendicular to the molecule. Owing to
the molecular site symmetry, only dz is non-zero. The molecular dipole moment from the AIM integrated properties and
the multipole model are equal within one s.u. even though the
individual atomic contributions are completely different. The
Ê correspond to 6.56 and 6.2 (5) D,
values 1.37 and 1.30 (11) e A
Acta Cryst. (2004). A60, 371±381
respectively. For the multipole dipole, the charge contribution
is generally small except for the O atom, which dominates the
total dipole moment. The net contribution of the NH2 group is
almost zero. This is in contrast to the AIM partitioning where
the NH2 groups contribute 66% of the total moment. These
differences are due to the completely different atomic charges.
The AIM molecular dipole moments from the periodic
RHF calculations are 6.87, 6.85 and 6.96 D for 6-21G(d,p),
6-31G(d,p) and 6-311G(d,p), respectively. The in¯uence of
geometry and/or computational precision can be appreciated
by comparison of the 6-31G(d,p) moments at the 123 and 12 K
SCM geometries: for the 12 K geometry, Gatti et al. (1994)
found 7.03 D. This variation re¯ects the level of precision that
can be expected from these calculations. In this view, the
agreement between the periodic RHF calculations and the
present experiment is good.
From X-ray diffuse scattering along the c axis of urea,
Lefebvre (1973) derived acoustic and optical phonon dispersion curves. By using a lattice dynamical model, the dipole
moment was estimated to be 4.66 D. Later, based on coherent
Henrik Birkedal et al.
Charge density of urea
379
research papers
inelastic neutron scattering on deuterated urea, Lefebvre et al.
(1975) established an improved force-constant model that
yielded a dipole moment of 3.99 D. Estimates of the solution
and gas-phase dipole moment are collected in Table S9. They
range from 3.83 to 6.38 D with a mean value of 5.1 D. Even for
measurements in the same solvent, variations are large. In
water at room temperature, the values range from 4.20 to
5.79 D with an average of 5.0 D. In view of the expected dipole
enhancement in the solid, the Lefebvre (1973) and Lefebvre et
al. (1975) values appear to be somewhat small. In Gatti et al.'s
(1994) RHF/6-31G(d,p) calculations, a clear enhancement was
seen: the molecular dipole moment in the bulk was found to be
larger by 1.9 D than the isolated molecule moment with the
same geometry. A similar, but somewhat smaller, enhancement of 1.41 D was found by multipole modelling of RHF/
DZPT structure factors by Spackman et al. (1999). If this
enhancement can be taken as valid, a value of about 6.5±7 D
would be expected in the solid state in perfect agreement with
the present values. Spackman et al. (1988) analysed the SCSS
multipole model and found a dipole moment of 5.4 (5) D ± not
inconsistent with the present results. ZSTVF obtained the
rather small value of 3.8 D.
3.4. Multipole refinement of a non-centrosymmetric system
The structure of urea is non-centrosymmetric and, with the
exception of symmetry-restricted phases, the structure-factor
phases can take any value between 0 and 2. Therefore, it can
be expected that modelling the charge density will be more
dif®cult than for centrosymmetric structures. In a series of
detailed studies on noise-free ab initio data sets, Spackman
and co-workers have shown that the multipole model is
capable of recovering phases (Spackman & Byrom, 1997) and
interaction densities (Spackman et al., 1999)6 faithfully ± at
least in well behaved systems like urea. de Vries et al. (2000)
investigated the in¯uence of noise on the modelling of the
interaction density with structure factors calculated with
periodic denssity functional theory (DFT) at the B-PWGGA/
6-21G(d,p) level. They calculated 412 structure factors
matching the data set of ZSTVF and included temperature
effects by applying the SCM 123 K ADPs on Hirshfeld
partitioned atomic densities. This data set was then used as
input for multipole modelling. The multipole model was rather
restricted: the core populations of C, O and N atoms were kept
®xed at 2, one parameter only was used for the valence shells
of the three heavy atoms and the radial functions of the H
atoms were kept ®xed. The re®nement included the hexadecapole level on C, N and O atoms and quadrupoles on H
atoms. This model recuperated the salient features of the
interaction density. They then proceeded by including noise in
the data by adding a random fraction of the ZSTVF s.u.'s to
each structure factor. The ensuing re®nement did not recover
the interaction density. Therefore the authors concluded that a
data set like ZSTVF is not of suf®cient quality for modelling
6
The interaction density is the difference between a theoretical molecular
density and the experimental one.
380
Henrik Birkedal et al.
Charge density of urea
the interaction density and thus the intermolecular interactions. They conclude by adding the caveat ` . . . an experiment
in which more structure factors are measured more accurately
would give better results'. It would appear that the present
experiment has achieved exactly this.
4. Conclusions
The present data set of urea is of unprecedented resolution
and precision. The re®ned nuclear parameters are in good
agreement with the SCM neutron results and additionally
show that non-negligible anharmonic contributions to the
thermal motion are present at N and C atoms. The features of
the derived static charge density ®t very well into the general
picture provided by high-precision studies of other systems. It
can thus safely be stated that the most precise electron density
of urea available to date has been extracted from the present
data set. This result could not have been achieved without the
use of synchrotron radiation and very high precision instrumentation.
Our results underline the need for improving the level of
theoretical calculations. The 6-31G and 6-311G series of basis
sets were found to be inadequate unless extensive sets of
polarization functions are added. These basis sets have a
common drawback: the valence s and p functions have a
common exponent. This was originally motivated by limited
computer power. For relatively small systems (including also
molecules larger than urea), this limitation is no longer justi®ed and other basis sets should be used. For comparison of
integrated properties, calculations must include correlation
effects. Note that these de®ciencies of current ab initio
calculations are particularly prominent in urea because of the
highly polarized CÐO bond. In less extreme systems, the
effects are smaller and more moderate levels of theory can be
expected to function well.
In view of the current active discussions on the choice of
radial basis functions, it would appear that studies like the
present one, where the focus is on extreme precision for a
small systems amenable to calculation, are of the utmost
importance. In addition, it would be advantageous if also the
average charge-density study would extend to higher angles
and higher precision thereby allowing more extensive
modelling. Finally, we recommend a more ¯exible attitude of
the charge-density community regarding the choice and
re®nement of radial basis functions.
We thank the staff of the Swiss±Norwegian Beam Lines for
their assistance and patience during the preparation and
execution of these measurements. Additional support from
the Danish Research Training Council and the Danish Natural
Sciences Research Council (HB) is gratefully acknowledged.
We thank Dr Claus Flensburg, Professor Mark A. Spackman,
Professor Robert F. Stewart and Professor Sine Larsen for
valuable discussions.
Acta Cryst. (2004). A60, 371±381
research papers
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